\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

Let \(F\colon M\to N\) smooth, \(\omega\in T^*N\) and \(u\colon N\to \mathbb{R}\) continuous, then

\begin{equation*} F^*(u\omega)=(u\circ F)F^*\omega. \end{equation*}

[@lee2013smooth_manifolds, Proposition 11.25]